Why is maths so difficult Anxiety, self-esteem and stuck thinking.

1. Why is maths so difficult? Anxiety, self-esteem and 'stuck' thinking. Jane WarrenDyslexia Tutor AssessorLearning Differences CentreUniversity of Southampton I feel a complete fraud speaking at this conference. At school I was always one of the maths avoiders, who was overjoyed to pass maths O level because I thought it meant I�d never have to do maths again. Many years later doing primary PGCE, of course I did have to do it again � and found it much more enjoyable second time around. Perhaps I was just a late developer. Then I found myself teaching as much maths as literacy in my role as SENCO. I always felt that numeracy was the �poor relation� in dyslexia training, so chose to examine the effects of maths anxiety on learners at KS2 as part of my MSc in SpLD. Clare Trott read this paper and invited me here today. Now, working in HE, I still find most of my colleagues reluctant to tackle the maths implications of dyslexia. This talk, therefore, aims to offer my perspective on the impact maths anxiety can have on learners at all levels, and what, if anything, can be done about it. I hope that a fruitful discussion may emerge following this talk. I�m not sure it matters whether we call these learners dyslexic, dyscalculic, maths anxious or lacking in confidence. Equally, maths anxiety, unlike literacy difficulty, does not seem particularly age or phase-specific. The important thing is to help the learners overcome their fear of maths. In many cases, we may first need to confront our own.I feel a complete fraud speaking at this conference. At school I was always one of the maths avoiders, who was overjoyed to pass maths O level because I thought it meant I�d never have to do maths again. Many years later doing primary PGCE, of course I did have to do it again � and found it much more enjoyable second time around. Perhaps I was just a late developer. Then I found myself teaching as much maths as literacy in my role as SENCO. I always felt that numeracy was the �poor relation� in dyslexia training, so chose to examine the effects of maths anxiety on learners at KS2 as part of my MSc in SpLD. Clare Trott read this paper and invited me here today. Now, working in HE, I still find most of my colleagues reluctant to tackle the maths implications of dyslexia. This talk, therefore, aims to offer my perspective on the impact maths anxiety can have on learners at all levels, and what, if anything, can be done about it. I hope that a fruitful discussion may emerge following this talk. I�m not sure it matters whether we call these learners dyslexic, dyscalculic, maths anxious or lacking in confidence. Equally, maths anxiety, unlike literacy difficulty, does not seem particularly age or phase-specific. The important thing is to help the learners overcome their fear of maths. In many cases, we may first need to confront our own.

2. A few statistics One of the principal difficulties seems to be the comparative respectability of maths incompetence. Literacy is a visible and fundamental skill � poor spelling and grammar are highly embarrassing and public perception associates them with low ability or lack of education � however misguided we in the dyslexia field know this to be. Poor mathematical ability, on the other hand, can almost be a matter for pride, whatever the person�s ability. (Sharma, Charles Clarke, and even Cohn in 1971 � so it isn�t calculators which are to blame.) Why might this be? Quote from Babbage. Let�s examine a few statistics, in order to help confront any fears we may have. These emerged from my original paper. 93% - US people unable to do algebra or multi-step operations. 12.6% - number of 2004 UK Year 11s who took maths GCSE. Astonishingly, this was still the single most commonly-taken GCSE of all subjects. 10% - percentage of the above cohort who gained grade E or below, thus left with the maths competence of an 11 year old. 22% - percentage of American adults who cannot do simple arithmetic 8% - percentage of American girls who do higher maths � and are thus able to enter high-earning careers in science and technology. 60% - proportion of dyslexics who also experience mathematical difficulties One of the principal difficulties seems to be the comparative respectability of maths incompetence. Literacy is a visible and fundamental skill � poor spelling and grammar are highly embarrassing and public perception associates them with low ability or lack of education � however misguided we in the dyslexia field know this to be. Poor mathematical ability, on the other hand, can almost be a matter for pride, whatever the person�s ability. (Sharma, Charles Clarke, and even Cohn in 1971 � so it isn�t calculators which are to blame.) Why might this be? Quote from Babbage. Let�s examine a few statistics, in order to help confront any fears we may have. These emerged from my original paper. 93% - US people unable to do algebra or multi-step operations. 12.6% - number of 2004 UK Year 11s who took maths GCSE. Astonishingly, this was still the single most commonly-taken GCSE of all subjects. 10% - percentage of the above cohort who gained grade E or below, thus left with the maths competence of an 11 year old. 22% - percentage of American adults who cannot do simple arithmetic 8% - percentage of American girls who do higher maths � and are thus able to enter high-earning careers in science and technology. 60% - proportion of dyslexics who also experience mathematical difficulties

3. Self-esteem or maths-esteem? Obviously all children learn maths. As with every subject, some learn it more easily than others do, but it very frequently does cause anxiety, as the previous slide suggested. What happens when it does? In the original study which gave rise to this lecture, eight children were interviewed. Four had dyslexia or other literacy difficulties, the other four had none, but all found maths deeply challenging. How might they respond to this? All the girls, dyslexic or non-dyslexic, attributed maths difficulty to a failure to listen to the teacher. This suggests that they had picked on route one. However, a subtext to their responses was frequently a dislike of the subject itself and a profound lack of understanding of its usefulness. This indicates that they were not far off route three as well. The boys were much more likely to attribute maths difficulty to an innate attribute of the brain. One very dyslexic boy, whose maths was actually much better than his literacy, when asked �What makes people good at maths� replied simply �just cleverer.� The most dyscalculic boy had a notion that everyone else could do it because they�d done it before. All these children were in my learning support group for maths but none were prepared to admit to themselves that they were poor at maths. A typical response was �I wouldn�t say I�m the best but I�m not bad.� This is in many ways reassuring but does suggest that route 2 is likely to be avoided in self-defence. If route 1 fails, therefore, route 3 may well be the next step. Low self-esteem is displaced by low maths esteem; the subject is dropped as early as possible and maths incompetence and anxiety in the adult population persists.Obviously all children learn maths. As with every subject, some learn it more easily than others do, but it very frequently does cause anxiety, as the previous slide suggested. What happens when it does? In the original study which gave rise to this lecture, eight children were interviewed. Four had dyslexia or other literacy difficulties, the other four had none, but all found maths deeply challenging. How might they respond to this? All the girls, dyslexic or non-dyslexic, attributed maths difficulty to a failure to listen to the teacher. This suggests that they had picked on route one. However, a subtext to their responses was frequently a dislike of the subject itself and a profound lack of understanding of its usefulness. This indicates that they were not far off route three as well. The boys were much more likely to attribute maths difficulty to an innate attribute of the brain. One very dyslexic boy, whose maths was actually much better than his literacy, when asked �What makes people good at maths� replied simply �just cleverer.� The most dyscalculic boy had a notion that everyone else could do it because they�d done it before. All these children were in my learning support group for maths but none were prepared to admit to themselves that they were poor at maths. A typical response was �I wouldn�t say I�m the best but I�m not bad.� This is in many ways reassuring but does suggest that route 2 is likely to be avoided in self-defence. If route 1 fails, therefore, route 3 may well be the next step. Low self-esteem is displaced by low maths esteem; the subject is dropped as early as possible and maths incompetence and anxiety in the adult population persists.

4. What is maths anxiety? How would you react if asked: What is 7 x 8? How much would something cost if the original price of �120 was discounted by 15%? Solve 3x + 2 = 5x - 5 � � � Is the answer <1 or >1? Read this passage aloud to the group How many people here would be anxious about these? The first one seems, at least in part, age dependent. Difficulty with learning times tables is a well-known dyslexic trait and so we enquire about it when screening students. Older, returning, students frequently do not report a difficulty: rote over-learning clearly had some effect! The second question is one that we also present, orally, to students at screening. They are invited to talk it through but not to write it down. Most would prefer to � they have memorised an algorithm but cannot take a mental, step by step approach. I should also add that several of our tutors do not ask it at all, this betraying their own maths anxiety. I have a confession to make about number three. I still experience considerable algebra anxiety so asked my Year 10 daughter for a good example. I doubt if I could solve it now. The fourth is fine if you understand fractions, but they are a known problem area which turns many people off maths. The last has been put in to underline the difference between the dyslexic and general populations. In my experience reading aloud, especially unrehearsed, is detested by most dyslexic people and is a very diagnostic question. Conversely, many non-dyslexic people may well find it preferable to tackling the maths!How many people here would be anxious about these? The first one seems, at least in part, age dependent. Difficulty with learning times tables is a well-known dyslexic trait and so we enquire about it when screening students. Older, returning, students frequently do not report a difficulty: rote over-learning clearly had some effect! The second question is one that we also present, orally, to students at screening. They are invited to talk it through but not to write it down. Most would prefer to � they have memorised an algorithm but cannot take a mental, step by step approach. I should also add that several of our tutors do not ask it at all, this betraying their own maths anxiety. I have a confession to make about number three. I still experience considerable algebra anxiety so asked my Year 10 daughter for a good example. I doubt if I could solve it now. The fourth is fine if you understand fractions, but they are a known problem area which turns many people off maths. The last has been put in to underline the difference between the dyslexic and general populations. In my experience reading aloud, especially unrehearsed, is detested by most dyslexic people and is a very diagnostic question. Conversely, many non-dyslexic people may well find it preferable to tackling the maths!

5. Modes of thinking Jan Robertson has posited three modes of mathematical thinking. Intuitive thinking involves �everyday� concrete maths in �real-life� contexts such as estimating the shopping bill, rounding money, working out how much paint is needed for a room, thinking about how high or far away things are. Many people do not have trouble with this but may still regard themselves as bad at maths. Perhaps they are unaware that theses activities are maths. Toolbox mode is made available by learning all those rules. The girls in my study who thought you could get good by listening to the teacher meant that you could get hold of the toolbox. Of course it is important to learn rules and algorithms but not at the expense of conceptual understanding. The snag with the toolbox is that some students think that that is all there is to maths. As one girl said in Sheila Tobia�s famous 1993 study: What I learned in ten years of school mathematics was what to do when I remembered what to do. What I really needed to know was what to do when I forgot. Abstract mode gives access to higher-order mathematics. Deductive reasoning, creative problem-solving, conceptual grasp and confidence to fail characterise this mode. Interestingly, many dyslexic people are strong in the abstract, conceptual mode. One of the boys in my study had a superb internal model of the number system which allowed him to retrieve number facts quickly despite a typical difficulty with memorising them. Skilled mathematicians will move between the modes as required. The maths-anxious person, in my experience, is likely to remain stuck in one of the three modes. A dyscalculic person may be unable to access any of them.Jan Robertson has posited three modes of mathematical thinking. Intuitive thinking involves �everyday� concrete maths in �real-life� contexts such as estimating the shopping bill, rounding money, working out how much paint is needed for a room, thinking about how high or far away things are. Many people do not have trouble with this but may still regard themselves as bad at maths. Perhaps they are unaware that theses activities are maths. Toolbox mode is made available by learning all those rules. The girls in my study who thought you could get good by listening to the teacher meant that you could get hold of the toolbox. Of course it is important to learn rules and algorithms but not at the expense of conceptual understanding. The snag with the toolbox is that some students think that that is all there is to maths. As one girl said in Sheila Tobia�s famous 1993 study: What I learned in ten years of school mathematics was what to do when I remembered what to do. What I really needed to know was what to do when I forgot. Abstract mode gives access to higher-order mathematics. Deductive reasoning, creative problem-solving, conceptual grasp and confidence to fail characterise this mode. Interestingly, many dyslexic people are strong in the abstract, conceptual mode. One of the boys in my study had a superb internal model of the number system which allowed him to retrieve number facts quickly despite a typical difficulty with memorising them. Skilled mathematicians will move between the modes as required. The maths-anxious person, in my experience, is likely to remain stuck in one of the three modes. A dyscalculic person may be unable to access any of them.

6. Getting stuck Stuck in the toolbox: �Maths is a set of rules and procedures.� How maths is often taught and perceived at school Stuck in intuitive mode: �I can give you a ballpark figure.� A common reaction in people not in education Stuck in abstract mode: �Algebra is easy; numbers get in the way.� Surprisingly common in the dyslexic population In my study one boy did show signs of developmental dyscalculia. Even using concrete materials such as Dienes he was unable to grasp subtraction. In individual sessions you could almost see the shutters come down in his head. He could not access any mode reliably and embodied the Utter Blank Look (UBL) error type (Jan again). Being stuck in toolbox mode implies that the rules may have been learnt at school but the underlying concepts poorly grasped. This is a problem at any stage of learning: it characterised all the non-dyslexic girls in my study, and is also the most common reason for students to attend the excellent courses here at the MLSC. Such learners believe that �Maths is a set of calculation procedures and techniques based on �rules� set up by some clever Maths person. The teacher shows you how to do it, then you practise and try to remember.� Error types include The Confident Wrong Answer (CWA). Intuitive mathematicians did not feature in my study, almost by definition. However, I am sure that I was able to use my own intuitive mathematical sense to support re-acquainting myself with the toolbox during my PGCE. In my case numbers are fine but algebra just does not compute. I can move comfortably between toolbox and intuitive modes but have trouble in the abstract. The boy in my study who had the internal schema of the number system but intensely disliked applied maths (increasingly emphasised in the Nat Curr) was at least partly stuck in abstract mode. Short-term memory difficulties associated with his dyslexia gave rise to Inordinate Length of Time (ILT) errors.In my study one boy did show signs of developmental dyscalculia. Even using concrete materials such as Dienes he was unable to grasp subtraction. In individual sessions you could almost see the shutters come down in his head. He could not access any mode reliably and embodied the Utter Blank Look (UBL) error type (Jan again). Being stuck in toolbox mode implies that the rules may have been learnt at school but the underlying concepts poorly grasped. This is a problem at any stage of learning: it characterised all the non-dyslexic girls in my study, and is also the most common reason for students to attend the excellent courses here at the MLSC. Such learners believe that �Maths is a set of calculation procedures and techniques based on �rules� set up by some clever Maths person. The teacher shows you how to do it, then you practise and try to remember.� Error types include The Confident Wrong Answer (CWA). Intuitive mathematicians did not feature in my study, almost by definition. However, I am sure that I was able to use my own intuitive mathematical sense to support re-acquainting myself with the toolbox during my PGCE. In my case numbers are fine but algebra just does not compute. I can move comfortably between toolbox and intuitive modes but have trouble in the abstract. The boy in my study who had the internal schema of the number system but intensely disliked applied maths (increasingly emphasised in the Nat Curr) was at least partly stuck in abstract mode. Short-term memory difficulties associated with his dyslexia gave rise to Inordinate Length of Time (ILT) errors.

7. Successful mathematics learning unites a number of factors. If any of the pieces are missing, maths will be hard. Mathematicians utilise flexible thinking, with the confidence to try things out and �have a go�. However, many researchers reiterate that maths anxiety is often characterised by the fear of making mistakes. This is partly because early school mathematics learning focuses strongly on the procedural at the expense of the conceptual, which perhaps encourages all learners to assume that all maths is solvable by the application of learned formulae. Even if students are thereby motivated to learn the formulae they will lose out on the pattern recognition and conceptual side: I would further suggest that if learning the formulae is extra difficult for any reason they will probably learn maths avoidance instead. Steve Chinn, for example, observed that what characterised the young dyslexic mathematician was the proportion of no attempts. Similarly, we put far too much emphasis on obtaining the right answer. It is far more important for me that students understand the concepts and reasoning leading up to it. However much I tried to emphasis this in teaching , the children in my study did not share this belief. All were terrified of giving or writing the wrong answer. I suspect this is true at all stages of education. Another false emphasis may be on �rapid� retrieval of number facts. This simply exacerbates anxiety, especially among dyslexic learners, when a measured approach may well be more appropriate as well as less threatening. The problem-solving aspects of maths are not best approached by the �snap answer� method. Mahesh Sharma further emphasises that Mathematics is a second language and should be taught as such. It is exclusively bound to the symbolic representation of ideas. The syntax, terminology, and the translation from English to math language and back must be directly and deliberately taught! He also suggests that concepts can and should be taught before number facts � this �algebra before arithmetic� approach is having an impact on US early maths teaching.Successful mathematics learning unites a number of factors. If any of the pieces are missing, maths will be hard. Mathematicians utilise flexible thinking, with the confidence to try things out and �have a go�. However, many researchers reiterate that maths anxiety is often characterised by the fear of making mistakes. This is partly because early school mathematics learning focuses strongly on the procedural at the expense of the conceptual, which perhaps encourages all learners to assume that all maths is solvable by the application of learned formulae. Even if students are thereby motivated to learn the formulae they will lose out on the pattern recognition and conceptual side: I would further suggest that if learning the formulae is extra difficult for any reason they will probably learn maths avoidance instead. Steve Chinn, for example, observed that what characterised the young dyslexic mathematician was the proportion of no attempts. Similarly, we put far too much emphasis on obtaining the right answer. It is far more important for me that students understand the concepts and reasoning leading up to it. However much I tried to emphasis this in teaching , the children in my study did not share this belief. All were terrified of giving or writing the wrong answer. I suspect this is true at all stages of education. Another false emphasis may be on �rapid� retrieval of number facts. This simply exacerbates anxiety, especially among dyslexic learners, when a measured approach may well be more appropriate as well as less threatening. The problem-solving aspects of maths are not best approached by the �snap answer� method. Mahesh Sharma further emphasises that Mathematics is a second language and should be taught as such. It is exclusively bound to the symbolic representation of ideas. The syntax, terminology, and the translation from English to math language and back must be directly and deliberately taught! He also suggests that concepts can and should be taught before number facts � this �algebra before arithmetic� approach is having an impact on US early maths teaching.

8. A way forward? Start from wherever the learner is Build motivation Emphasise process not product Make time and space available Talk through anxieties in order to build self-esteem Spend time on the language of maths Finally, can we cut through the thickets of anxiety and misconception to increase confidence in maths? Too much conventional maths teaching seems to pile up facts and concepts without ensuring that principles are grasped. At whatever level, it is vital to start from where the learner is without preconceptions of where they should be. At HE level, it is likely that motivation may be mainly extrinsic � maths is part of many courses such as nursing and psychology which may not have been chosen for their mathematical content. Lack of practice may seem to translate as lack of aptitude and sensitive support is needed to help students through this stage. Building self-esteem is vital in building maths esteem. Making time and space available will help � time to avoid the �instant response� mentality of the maths classroom � emotional space to talk the problems through. Sheila Tobias� study found that a weekly session spent talking through the difficulties was more productive in raising confidence and attainment than a similar time spent on �remedial� maths. Explicit teaching of the language of maths Finally, can we cut through the thickets of anxiety and misconception to increase confidence in maths? Too much conventional maths teaching seems to pile up facts and concepts without ensuring that principles are grasped. At whatever level, it is vital to start from where the learner is without preconceptions of where they should be. At HE level, it is likely that motivation may be mainly extrinsic � maths is part of many courses such as nursing and psychology which may not have been chosen for their mathematical content. Lack of practice may seem to translate as lack of aptitude and sensitive support is needed to help students through this stage. Building self-esteem is vital in building maths esteem. Making time and space available will help � time to avoid the �instant response� mentality of the maths classroom � emotional space to talk the problems through. Sheila Tobias� study found that a weekly session spent talking through the difficulties was more productive in raising confidence and attainment than a similar time spent on �remedial� maths. Explicit teaching of the language of maths

9. References Benson, I. (2007) �Ditching Piaget� in Prospect, 130, pp 16-17 Chinn, S. and Ashcroft, J. (1998) Mathematics for dyslexics: a teaching handbook, (2nd. ed.) London, Whurr Cockroft, W. (1982) Mathematics Counts, London, HMSO Cohn, R. (1971) �Arithmetic and learning disabilities� in Myklebust, M.(Ed.) Progress In learning disabilities,11, New York: Grunt & Stratton Robertson, J. (2005) �Does Dyscalculia affect the Learning of Mathematical Concepts? (The� Twoness� of Two)� in Brain-HE Conference Proceedings 2004-5 http://brainhe.com/resources/ Robertson, J. and Wright, F. (2005) �Learning support for students with mathematical difficulties� in in Brain-HE Conference Proceedings 2004-5 http://brainhe.com/resources/ Snyder, T., Hoffman, C., Geddes, C. (1997) Digest of Education Statistics 1997, NCES 98-015. U.S. Department of Education http://nces.gov/pubs/digest97/98015.html Tobias, S. (1993) Overcoming Math Anxiety, New York, Norton Trivett, J. (1959) �The coloured sticks� in New Scientist, 5, 12, pp 1183-1186 http://parents.sociality.tv/about/socialityohs.pdf